# Dab solver - Parseval's theorem

./dab_solver.py -page:
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More generally, given an abelian [[topological group]] G with [[Pontryagin dual]] G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled  measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line R, G^ is also R and the unitary transform is the [[Fourier transform]] on the real line. When G is the [[cyclic group]] Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called [[discrete-time Fourier transform]] in applied contexts.

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